Geometry and Group Theory

Transcription

1 Geometry and Group Theory ABSTRACT In this course, we develop the basic notions of Manifolds and Geometry, with applications in physics, and also we develop the basic notions of the theory of Lie Groups, and their applications in physics.

2 Contents 1 Manifolds Some set-theoretic concepts Topological Spaces Manifolds Tangent vectors Co-vectors An interlude on vector spaces and tensor products Tensors The Metric Tensor Covariant differentiation The Riemann curvature tensor Differential Forms Integration, and Stokes Theorem The Levi-Civita Tensor and Hodge Dualisation The δ Operator and the Laplacian Spin connection and curvature 2-forms General Relativity; Einstein s Theory of Gravitation The Equivalence Principle A Newtonian Interlude The Geodesic Equation The Einstein Field Equation The Schwarzschild Solution Orbits Around a Star or Black Hole Lie Groups and Algebras Definition of a Group Lie Groups The Classical Groups Lie Algebras Roots and Weights Root Systems for the Classical Algebras

3 The material in this course is intended to be more or less self contained. However, here is a list of some books and other reference sources that may be helpful for some parts of the course: 1. J.G. Hocking and G.S. Young, Topology, (Addison-Wesley, 1961). This is a very mathematical book on topological spaces, point-set topology, and some more advanced topics in algebraic topology. (Not for the faint-hearted!) 2. T. Eguchi, P.B. Gilkey and A.J. Hanson. Gravitation, Gauge Theories and Differential Geometry, Physics Reports, 66, 213 (1980). This is a very readable exposition of the basic ideas, aimed at physicists. Some portions of this course are based fairly extensively on this article. It also has the merit that it is freely available for downloading from the web, as a PDF file. Go to type find a gilkey and a hanson, and follow the link to Science Direct for this article. Note that Science Direct is a subscription service, and you must be connecting from a URL in the tamu.edu domain, in order to get free access. 3. H. Georgi, Lie Algebras and Particle Physics, Perseus Books Group; 2nd edition (September 1, 1999). This is quite a useful introduction to some of the basics of Lie algebras and Lie groups, written by a physicist for physicists. It is a bit idiosyncratic in its coverage, but what it does cover is explained reasonably well. 4. R. Gilmore, Lie Groups Lie Algebras and Some of Their Applications, John Wiley & Sons, Inc (1974). A more complete treatment of the subject. Quite helpful, especially as a reference work. 2

4 1 Manifolds One of the most fundamental constructs in geometry is the notion of a Manifold. A manifold is, in colloquial language, the arena where things happen. Familiar examples are the threedimensional space that we inhabit and experience in everyday life; the surface of a ball, viewed as a two-dimensional closed surface on which, for example, an ant may walk; and the four-dimensional Minkowski spacetime that is the arena where special relativity may be formulated. In order to give a reasonably precise description of a manifold, it is helpful first to give a few rather formal definitions. It is not the intention in this course to make everything too formal and rigorous, so we shall try to strike a balance between formality and practical utility as we proceed. In particular, if things seem to be getting too abstract and rigourous at any stage, there is no need to panic, because it will probably just be a brief interlude before returning to a more intuitive and informal discussion. In this spirit, let us begin with some formal definitions. 1.1 Some set-theoretic concepts A set is a collection of objects, or elements; typically, for us, these objects will be points in a manifold. A set U is a subset of a set V if every element of U is also an element of V. If there exist elements in V that are not in the subset U, then U is called a proper subset of V. If U is a subset of V then the complement of U in V, denoted by V U, is the set of all elements of V that are not in U. If U is a subset but not a proper subset, then V U contains no elements at all. This set containing no elements is called the empty set, and is denoted by. By definition, therefore, is a subset of every set. The notion of the complement can be extended to define the difference of sets V and U, even when U is not a subset of V. Thus we have V \U = {x : x V and x / U}. (1.1) If U is a subset of V then this reduces to the complement defined previously. Two sets U and V are equal, U = V, if every element of V is an element of U, and vice versa. This is equivalent to the statement that U is a subset of V and V is a subset of U. From two sets U and V we can form the union, denoted by U V, which is the set of all elements that are in U or in V. The intersection, denoted by U V, is the set of all elements that are in U and in V. The two sets U and V are said to be disjoint if U V =, i.e. they have no elements in common. 3

5 Some straightforwardly-established properties are: A B = B A, A B = B A, A (B C) = (A B) C, A (B C) = (A B) C, (1.2) A (B C) = (A B) (A C), A (B C) = (A B) (A C). If A and B are subsets of C, then C (C A) = A, C (C B) = B, A (A\B) = A B, C (A B) = (C A) (C B), C (A B) = (C A) (C B). (1.3) 1.2 Topological Spaces Before being able to define a manifold, we need to introduce the notion of a topological space. This can be defined as a point set S, with open subsets O i, for which the following properties hold: 1. The union of any number of open subsets is an open set. 2. The intersection of a finite number of open subsets is an open set. 3. Both S itself, and the empty set, are open. It will be observed that the notion of an open set is rather important here. Essentially, a set X is open if every point x inside X has points round it that are also in X. In other words, every point in an open set has the property that you can wiggle it around a little and it is still inside X. Consider, for example, the set of all real numbers r in the interval 0 < r < 1. This is called an open interval, and is denoted by (0, 1). As its name implies, the open interval defines an open set. Indeed, we can see that for any real number r satisfying 0 < r < 1, we can always find real numbers bigger than r, and smaller than r that still themselves lie in the open interval (0, 1). By contrast, the interval 0 < r 1 is not open; the point r = 1 lies inside the set, but if it is wiggled to the right by any amount, no matter how tiny, it takes us to a point with r > 1, which is not inside the set. Given the collection {O i } of open subsets of S, we can define the notion of the limit point of a subset, as follows. A point p is a limit point of a subset X of S provided that 4

6 every open set containing p also contains a point in X that is distinct from p. This definition yields a topology for S, and with this topology, S is called a Topological Space. Some further concepts need to be introduced. First, we define a basis for the topology of S as some subset of all possible open sets in S, such that by taking intersections and unions of the members of the subset, we can generate all possible open subsets in S. An open cover {U i } of S is a collection of open sets such that every point p in S is contained in at least one of the U i. The topological space S is said to be compact if every open covering {U i } has a finite sub-collection {U i1,, U in } that also covers S. Finally, we may define the notion of a Hausdorff Space. The topological space S is said to obey the Hausdorff axiom, and hence to be an Hausdorff Space, if, for any pair of distinct points p 1 and p 2 in S, there exist disjoints open sets O 1 amd O 2, each containing just one of the two points. In other words, for any distinct pair of points p 1 and p 2, we can find a small open set around each point such that the two open sets do not overlap. 1 We are now in a position to move on to the definition of a manifold. 1.3 Manifolds Before giving a formal definition of a manifold, it is useful to introduce what we will recognise shortly as some very simple basic examples. First of all, consider the real line, running from minus to plus infinity. Slightly more precisely, we consider the open interval (, ), i.e. the set of points x such that < x <. We denote this by the symbol IR (the letter R representing the real numbers). In fact IR is the prototype example of a manifold; it is a one-dimensional topological space parameterised by the points on the real line. A simple extension of the above is to consider the n-dimensional space consisting of n copies of the real line. We denote this by IR n. A familiar example is three-dimensional Euclidean space, with Cartesian coordinates (x, y, z). Thus our familiar three-dimensional space can be called the 3-manifold IR 3. We can now give a formal definition of a smooth n-manifold, with a smooth atlas of charts, as 1. A topological space S 2. An open cover {U i }, which are known as patches 1 The great mathematician and geometer Michael Atiyah gave a nice colloquial definition: A topological space is Hausdorff if the points can be housed off. One should imagine this being spoken in a rather plummy English accent, in which the word off is pronounced orff. 5

7 3. A set (called an atlas) of maps φ i : U i IR n called charts, which define a 1-1 relation between points in U i and points in an open ball in IR n, such that 4. If two patches U 1 and U 2 intersect, then both φ 1 φ 1 2 and φ 2 φ 1 1 are smooth maps from IR n to IR n. The set-up described here will be referred to as an n-dimensional manifold M. Sometimes we shall use a superscript or subscript n, and write M n or M n. What does all this mean? The idea is the following. We consider a topological space S, and divide it up into patches. We choose enough patches so that the whole of S is covered, with overlaps between neighbouring patches. In any patch, say U 1, we can establish a rule, known as a mapping, between points in the patch and points in some open connected region (called an open ball) in IR n. This mapping will be chosen such that it is 1-1 (one to one), meaning that there is a unique invertible relation that associates to each point in U 1 a unique point in the open ball in IR n, and vice versa. We denote this mapping by φ 1. This is the notion of choosing coordinates on the patch U 1. See Figure 1. U 1 φ 1 R n M Figure 1: The map φ 1 takes points in U 1 into an open ball in IR n Now consider another patch U 2, which has some overlap with U 1. For points in U 2 we make another mapping, denoted by φ 2, which again gives a 1-1 association with points in an open ball in IR n. Now, consider points in the topological space S that lie in the intersection of U 1 and U 2. For such points, we have therefore got two different rules for mapping into a copy of IR n : we have the mapping φ 1, and the mapping φ 2. We are therefore in a position to go back and forth between the two copies of IR n. Note that we can do this because each of φ 1 and φ 2 was, by definition, an invertible map. We can start from a point in the open ball in the second copy of IR n, and then apply the inverse of the mapping φ 2, which we denote by φ 1 2, to take us back to a point in the patch U 2 that is also in U 1. Then, we apply the map φ 1 to take us forward to the open ball 6

8 in the first copy of IR n. The composition of these two operations is denoted by φ 1 φ 1 2. Alternatively, we can go in the other order and start from a point in the open ball of the first copy of IR n that maps back using φ 1 1 to a point in U 1 that is also in U 2. Then, we apply φ 2 to take us into the second copy of IR n. Going in this direction, the whole procedure is therefore denoted by φ 2 φ 1 1. See Figure 2. U 1 φ 1 R n M U 2 φ 2 φ 2 ο φ 1 1 R n Figure 2: φ 2 φ 1 1 maps IR n into IR n for points in the intersection U 1 U 2 Whichever way we go, the net effect is that we are mapping between a point in one copy of IR n and a point in another copy of IR n. Suppose that we choose coordinates (x 1, x 2,, x n ) on the first copy, and coordinates ( x 1, x 2,, x n ) on the second copy. Collectively, we can denote these by x i, and x i, where 1 i n. In the first case, the composition φ 1 φ 1 2 is therefore giving us an expression for the x i as functions of the x j. In the second case, φ 2 φ 1 1 is giving us x i as functions of the x j. So far, we have discussed this just for a specific point that lies in the intersection of U 1 and U 2. But since we are dealing with open sets, we can always wiggle the point around somewhat, and thus consider an open set of points within the intersection U 1 U 2. Thus our functions x i = x i ( x j ) and x i = x i (x j ) can be considered for a range of x i and x i values. This allows us to ask the question of whether the functions are smooth or not; in other words, are the x i differentiable functions of the x j, and vice versa? Thus we are led to the notion of a Differentiable Manifold, as being a manifold where the coordinates covering any pair of overlapping patches are smooth, differentiable functions of one another. One can, of course, consider different degrees of differentiability; in practice, we shall tend to assume that everything is C differentiable, meaning that we can differentiate infinitely many times. The functions that describe how the x i depend on the x j, or how the x i depend on the x j, are called the transition functions in the overlap region. Two atlases are said to be compatible if, wherever there are overlaps, the transition 7

9 functions are smooth. It is worth emphasising at this point that all this talk about multiple patches is not purely academic. The reason why we have been emphasising this issue is that in general we need more than one coordinate patch to cover the whole manifold. To illustrate this point, it is helpful to consider some examples The circle; S 1 We have already met the example of the real line itself, as the one-dimensional manifold IR. This manifold can be covered by a single coordinate patch, namely we just use the coordinate x, < x <. There is another example of a one-dimensional manifold that we can consider, namely the circle, denoted by S 1. We can think of the circle as a real line interval, where the right-hand end of the line is identified with the left-hand end. Thus, for the unit circle, we can take a coodinate interval 0 θ < 2π, with the rule that the point θ = 2π is identified with the point θ = 0. However, θ is not a good coordinate everywhere on the circle, because it has a discontinuity where it suddently jumps from 2π to 0. To cover the circle properly, we need to use (at least) two coordinate patches. To see how this works, it is convenient to think of the standard unit circle x 2 + y 2 = 1 centred on the origin in the (x, y) plane, and to consider the standard polar angular coordinate θ running counter-clockwise around the circle. However, we shall only use θ to describe points on the circle corresponding to 0 < θ < 2π. Call this patch, or set of points, U 1. Introduce also another angular coordinate, called θ, which starts from θ = 0 (more precisely, we shall consider only θ > 0, not allowing θ = 0 itself) over on the left-hand side at θ = π, and runs around counter-clockwise until it (almost) returns to its starting point as θ approaches 2π. We shall use θ only in the interval 0 < θ < 2π. This patch of S 1 will be called U 2. Thus we have the patch U 1, which covers all points on S 1 except (x, y) = (1, 0), and the patch U 2, which covers all points on S 1 except (x, y) = ( 1, 0). The intersection of U 1 and U 2 therefore comprises all points on S 1 except for the two just mentioned. It therefore comprises two disconnected open intervals, one consisting of points on S 1 that lie above the x axis, and the other comprising points on S 1 that lie below the x axis. We may denote these two intervals by (U 1 U 2 ) + and (U 1 U 2 ) respectively. See Figure 3. The map φ 1 from points in U 1 into IR is very simple: we have chosen just to use θ, lying in the open interval 0 < θ < 2π. For U 2, we have the map φ 2 into the open interval 0 < θ < 2π in IR. The relation between the two coordinates in the overlap region defines 8

10 S 1 θ=2π θ=0 θ=0 θ=2π Figure 3: The coordinates θ and θ cover the two patches of S 1 the transition functions. Since we have an overlap region comprising two disconnected open intervals, we have to define the transition functions in each interval. This can be done easily by inspection, and we have (U 1 U 2 ) + : θ = θ + π (U 1 U 2 ) : θ = θ π. (1.4) It is obvious, in this example, that the transition functions are infinitely differentiable The 2-sphere; S 2 For a second example, consider the 2-sphere, denoted by S 2. We can think of this as the surface of the unit ball in Euclidean 3-space. Thus, if we introduce coordinates (x, y, z) on Euclidean 3-space IR 3, we define S 2 as the surface x 2 + y 2 + z 2 = 1. (1.5) We can think of using the spherical polar coordinates (θ, φ) on S 2, defined in the standard way: x = sin θ cos φ, y = sin θ sin φ, z = cos θ. (1.6) However, these coordinates break down at the north pole N, and at the south pole S, since at these points θ = 0 and θ = π there is no unique assignment of a value of φ. Instead, we can introduce stereographic coordinates, and define two charts: 9

11 For a point P on the sphere, take the straight line in IR 3 that starts at the north pole N, passes through P, and then intersects the z = 0 plane at (x, y). A simple geometric calculation shows that if the point P has spherical polar coordinates (θ, φ), then the corresponding point of intersection in the z = 0 plane is at x = cot 1 2 θ cos φ, y = cot 1 2θ sin φ. (1.7) This mapping from points in S 2 into points in IR 2 works well except for the point N itself: the north pole gets mapped out to infinity in the (x, y) plane. Let us call U the patch of S 2 comprising all points except the north pole N. We can get a well-defined mapping for a second patch U +, consisting of all points in S 2 except the south pole S, by making an analogous stereographic mapping from the south pole instead. A simple geometric calculation shows that the straight line in IR 3 joining the south pole to the point on S 2 parameterised by (θ, φ) intersects the z = 0 plane at x = tan 1 2 θ cos φ, ỹ = tan 1 2θ sin φ. (1.8) Thus we have a mapping given by (1.7) from U into IR 2, with coordinates (x, y), and a mapping given by (1.8) from U + into IR 2, with coordinates ( x, ỹ). In the intersection U U +, which comprises all points in S 2 except the north and south poles, we can look at the relation between the corresponding coordinates. From (1.7) and (1.8), a simple calculation shows that we have x = x x 2 + y 2, ỹ = y x 2 + y 2. (1.9) Conversely, we may express the untilded coordinates in terms of the tilded coordinates, finding x = x x 2 + ỹ 2, y = ỹ x 2 + ỹ 2. (1.10) It is easy to see that these transition functions defining the relations between the tilded and the untilded coordinates are infinitely differentiable, provided that x 2 + y 2 is not equal to zero or infinity. In other words, the transition functions are infinitely differentiable provided we omit the north and south poles; i.e., they are infinitely differentiable everywhere in the overlap of the two patches. The construction we have just described has provided us with an atlas comprising two charts. Clearly there was nothing particularly special about the way we chose to do this, except that we made sure that our atlas was big enough to provide a complete covering of S 2. We could, for example, add some more charts by repeating the previous discussion for 10

12 pairs of charts obtained by stereographic projection from (x, y, z) = (1, 0, 0) and ( 1, 0, 0), and from (0, 1, 0) and (0, 1, 0) as well. We would then in total have a collection of six charts in our atlas. A crucial point, though, which was appreciated even in the early days of map-making, is that you cannot cover the whole of S 2 with a single chart Incompatible Atlases It is not necessarily the case that the charts in one atlas are compatible with the charts in another atlas. A simple example illustrating this can be given by considering the onedimensional manifold IR. We have already noted that this can be covered by a single chart. Let us take z to represent the real numbers on the interval < z <. We can choose a chart given by the mapping φ : x = z. (1.11) We can also choose another chart, defined by the mapping φ : x = z 1/3. (1.12) Over the reals, each mapping gives a 1-1 relation between points z in the original manifold IR, and points in the copies of IR coordinatised by x or x respectively. However, these two charts are not compatible everywhere, since we have the relation x = x 1/3, which is not differentiable at x = Non-Hausdorff manifolds In practice we shall not be concerned with non-hausdorff manifolds, but is is useful to give an example of one, since this will illustrate that they are rather bizarre, and hence do not usually arise in situations of physical interest. Consider the following one-dimensional manifold. We take the real lines y = 0 and y = 1 in the (x, y) plane IR 2. Thus we have the lines (x, 0) and (x, 1). Now, we identify the two lines for all points x > 0. Thus we have a one-dimensional manifold consisting of two lines for x 0, which join together to make one line for x > 0. Now, consider the two points (0, 0) and (0, 1). These two points are distinct, since we are only making the identification of (x, 0) and (x, 1) for points where x is strictly positive. However, any open neighbourhood of (0, 0) necessarily intersects any open neighbourhood of (0, 1), since slightly to the right of x = 0 for any x, no matter how small, the two lines are identified. Thus, in Atiyah s words, the points (0, 0) and (0, 1) cannot be housed off into separate disjoint subsets. The only one-dimensional Hausdorff manifolds are IR and S 1. 11

13 1.3.5 Compact vs. non-compact manifolds When discussing topological spaces, we gave the definition of a compact topological space S as one for which every open covering {U i } has a finite sub-collection {U i1,, U in } that also covers S. The key point in this definition is the word every. To illustrate this, let us consider as examples the two simple one-dimensional manifolds that we have encountered; IR and S 1. As we shall see, IR is non-compact, whilst S 1 is compact. First, consider IR. Of course we can actually just use a single open set to over the whole manifold in this case, since if it is parameterised by the real number z, we just need to take the single open set < z <. Clearly if we took this as our open covering U then there exists a finite sub-collection (namely U itself no one said the sub-collection has to be a proper sub-collection) which also covers IR. However, we can instead choose another open covering as follows. Let U j be the open set defined by j 1 2 < z < j Thus U j describes an open interval of length just less than 2. Clearly {U j } for all integers j provides us with an open covering for IR, since each adjacent pair U j and U j+1 overlap. However, it is impossible to choose a finite subset of the U j that still provides an open cover of IR. By exhibiting an open covering for which a finite sub-collection does not provide an open covering of the manifold, we have proved that IR is not compact. A manifold that is not compact is called, not surprisingly, non-compact. Now, consider instead the manifold S 1. We saw in section (1.3.1) that we can cover S 1 with a minimum of two open sets. We could, of course, use more, for example by covering the circumference of the circle in short little sections of overlapping open sets. However, no matter how short we take the intervals, they must always have a non-zero length, and so after laying a finite number around the circle, we will have covered it all. We could choose an infinity of open sets that covered S 1, for example by choosing intervals of length 1 (in the sense 0 < z < 1) distributed around the circumference according to the rule that each sucessive interval starts at a point where the angle θ has advanced by 1 2 relative to the start of the previous interval. Since the circumference of the circle is traversed by advancing θ by 2π, it follows from the fact that π is transcendental that none of these unit intervals will exactly overlap another. However, it will be the case that we can choose a finite subset of these open intervals that is already sufficient to provide an open cover. No matter what one tries, one will always find that a finite collection of any infinite number of open sets covering S 1 will suffice to provide an open cover. Thus the manifold S 1 is compact. Of the other examples that we have met so far, all the manifolds IR n are non-compact, 12

14 and the manifold S 2 is compact Functions on manifolds A real function f on a manifold M is a mapping f : M IR (1.13) that gives a real number for each point p in M. If for some open set U in M we have a coordinate chart φ such that U is mapped by φ into IR n, then we have a mapping f φ 1 : IR n IR. (1.14) If the coordinates in IR n are called x i, then the mapping (1.14) can be written simply as f(x i ). In colloquial language we can say that f(x i ) represents the value of f at the point in M specified by the coordinates x i. In other words, now that it is understood that we may well need different coordinate patches to cover different regions of the manifold, we can usually just think of the chosen coordinates in some patch as being coordinates on the manifold, and proceed without explicitly reciting the full rigmarole about the mapping φ from M into IR n. The function f(x i ) is called a smooth function if it is a differentiable function of the coordinates x i in the patch where they are valid Orientable manifolds A manifold is said to be orientable if it admits an atlas such that in all overlapping regions between charts, the Jacobian of the relation between the coordinate systems satisfies 1.4 Tangent vectors ( x i ) det x j > 0. (1.15) We now turn to a discussion of vectors, and tensors, on manifolds. We should begin this discussion by forgetting certain things about vectors that we learned in kindergarten. There, the concept of a vector was introduced through the notion of the position vector, which was an arrow joining a point A to some other point B in three-dimensional Euclidean space. This is fine if one is only going to talk about Euclidean space in Cartesian coordinates, but it is not a valid way describing a vector in general. If the space is curved, such as the sphere, or even if it is flat but described in non-cartesian 13

15 coordinates, such as Euclidean 3-space described in spherical polar coordinates, the notion of a vector as a line joining two distant points A and B breaks down. What we can do is take the infinitesimal limit of this notion, and consider the line joining two points A and A + δa. In fact what this means is that we think of the tangent plane at a point in the space, and imagine vectors in terms of infinitesimal displacements in this plane. To make the thinking a bit more concrete, consider a 2-sphere, such as the surface of the earth. A line drawn between Ney York and Los Angeles is not a vector; for example, it would not make sense to consider the sum of the line from New York to Los Angeles and the line from Los Angeles to Tokyo, and expect it to satisfy any meaningful addition rules. However, we can place a small flat sheet on the surface of the earth at any desired point, and draw very short arrows in the plane of the sheet; these are tangent vectors at that particular point on the earth. The concept of a vector as an infinitesimal displacement makes it sound very like the derivative operator, and indeed this is exactly what a vector is. Suppose we consider some patch U in the manifold M, for which we introduce local coordinates x i in the usual way. Now consider a path passing through U, which may therefore be described by specifying the values of the coordinates of points along the path. We can do this by introducing a parameter λ that increases monotonically along the path, and so points in M along the path are specified by x i = x i (λ). (1.16) Consider now a smooth function f defined on M. The values of f at points along the path are therefore given by f(x i (λ)). By the chain rule, we shall have df n dλ = f dx i (λ) x i dλ, i=1 = f dx i (λ) x i dλ (1.17) Note that here, and throughout this course, we shall be using the Einstein summation convention, as is done in the second line, in which the summation over an index that appears exactly twice is understood. We may define the directed derivative operator along the path by which is a map taking smooth functions to IR: V d dλ, (1.18) f V f = df dλ 14 (1.19)

16 This obeys the linearity property V (f + g) = V f + V g (1.20) for any pair of smooth functions f and g, and also the Leibnitz property V (fg) = (V f)g + f(v g). (1.21) Such a map is called a tangent vector at the point p where the evaluation is made. If we have two different tangent vectors at the point p (corresponding to directional derivatives along two different curves that intersect at p), let us call them V = d/dλ and Ṽ = d/d λ, then linearity means that we shall have (V + Ṽ )f = V f + Ṽ f. (1.22) We can also multiply tangent vectors by constants and they are again tangent vectors. Thus the space of tangent vectors at a point p M is a vector space, which is called the Tangent Space at p, and denoted by T p (M). Its dimension is n, the dimension of the manifold M. This can be seen by considering Taylor s theorem in the local coordinate system x i : f(x) = f(x p ) + h i f x i +, (1.23) where h i x i x i p and xi p denotes the coordinates corresponding to the point p. Therefore if we define then we shall have V i V x i = dxi dλ, (1.24) V f = V i f x i, (1.25) and so we can take / x i as a basis for the vector space of tangent vectors at the point p. This shows that the dimension of the tangent vector space is equal to the number of coordinates x i, which is in turn equal to the dimension n of the manifold M. In order to abbreviate the writing, we shall commonly write to denote the tangent vector basis. i To summarise, we can write the tangent vector V = d/dλ as x i (1.26) V = V i i, (1.27) 15

17 where V i are the components of the vector V with respect to the basis i ; V i = dxi (λ) dλ. (1.28) (Of course here we are using the Einstein summation convention that any dummy index, which occurs twice in a term, is understood to be summed over the range of the index.) Notice that there is another significant change in viewpoint here in comparison to the kindergarten notion of a vector. We make a clear distinction betwen the vector itself, which is the geometrical object V defined quite independently of any coordinate system by (1.18), and its components V i, which are coordinate-dependent. 2 Indeed, if we imagine now changing to a different set of coordinates x i in the space, related to the original ones by x i = x i (x j ), then we can use the chain rule to convert between the two bases: V = V j x j = V j x i x j x i V i x i. (1.29) In the last step we are, by definition, taking V i to be the components of the vector V with respect to the primed coordinate basis. Thus we have the rule V i = x i x j V j, (1.30) which tells us how to transform the components of the vector V between the primed and the unprimed coordinate system. This is the fundamental defining rule for how a vector must transform under arbitrary coordinate transformations. Such transformations are called General Coordinate Transformations. Let us return to the point alluded to previously, about the vector as a linear differential operator. We have indeed been writing vectors as derivative operators, so let s see why that is very natural. Suppose we have a smooth function f defined on M. As we discussed before, we can view this, in a particular patch, as being a function f(x i ) of the local coordinates we are using in that patch. It is also convenient to suppress the i index on the coordinates x i in the argument here, and just write f(x). Now, if we wish to evaluate f at a nearby point x i + ξ i, where ξ i is infinitesimal, we can just make a Taylor expansion: f(x + ξ) = f(x) + ξ i i f(x) +, (1.31) 2 However, it sometimes becomes cumbersome to use the longer form of words the vector whose components are V i, and so we shall sometimes slip into the way of speaking of the vector V i. One should remember, however, that this is a slightly sloppy way of speaking, and the more precise distinction between the vector and its components should always be borne in mind. 16

18 and we can neglect the higher terms since ξ is assumed to be infinitesimal. Thus we see that the change in f is given by δf(x) f(x + ξ) f(x) = ξ i i f(x), (1.32) and that the operator that is implementing the translation of f(x) is exactly what we earlier called a vector field, ξ i i, (1.33) where δx i (x i + ξ i ) x i = ξ i. (1.34) Having defined T p (M), the tangent space at the point p M, we can then define the so-called tangent bundle as the space of all possible tangent vectors at all possible points: T (M) = p M T p (M). (1.35) This space is a manifold of dimension 2n, since to specify a point in it one must specify the n coordinates of M and also an n-dimensional set of basis tangent vectors at that point. It is sometimes called the velocity space, since it is described by a specification of the positions and the velocities / x i Non-coordinate bases for the tangent space In the discussion above, we have noted that i / x i forms a basis for the tangent space T p (M) at a point p in M. This is called a coordinate basis. We can choose to use different bases; any choice of n basis vectors that span the vector space is equally valid. Thus we may introduce quantities E i a, where 1 a n (and, as usual, 1 i n), and take our n basis vectors to be E a = E i a i. (1.36) As long as we have det(e i a ) 0, this basis will span the tangent space. Note that E i a need not be the same at each point in M; we can allow it to depend upon the local coordinates x i : E a = E i a (x) i. (1.37) A common terminology is to refer to E i a as the inverse vielbein (we shall meet the vielbein itself a little later). The coordinate index i is commonly also called a world index, while the index a is commonly called a tangent space index. 17

19 In addition to the general coordinate transformations x i x i = x i (x) that we discussed previously, we can also now make transformations on the tangent space index. In other words, we can make transformations from one choice of non-coordinate basis E i a to another, say E a i. This transformation can itself be different at different points in M: E a E a = Λ a b (x) E b. (1.38) Note that if we have a vector V = V i i, where V i are its components in the coordinate basis i, we can also write it as V = V a E a, (1.39) where V a denotes the tangent-space components of V with respect to the basis E a. Since V itself is independent of the coice of basis, it follows that the components V a must transform in the inverse fashion to the transformation (1.38) of E a, which we write as V a V a = Λ a b (x) V b, (1.40) where Λ a b Λ c b = δ a c. (1.41) It is straightforward to see that (1.38) and (1.40), together with (1.41), implies that V given in (1.39) is invariant under these local tangent-space transformations. In matrix notation, we can associate Λ a b with the matrix Λ, whose rows are labelled by a, and columns by b. Then from (1.41) we have that Λ b a corresponds to the inverse, Λ 1. If we view the set of n basis vectors E a as a row vector denoted by E, and the set of tangent-space components V a as a column vector denoted by V, then (1.38) and (1.40) can be written as E = E Λ 1, V = Λ V. (1.42) 1.5 Co-vectors We have so far met the concept of vectors V, which can be expanded in a coordinate basis i or a general tangent-space basis E a : V = V i i = V a E a. For every vector space X, there exists the notion of its dual space X, which is the space of linear maps X : X IR. (1.43) What this means is that if V is any vector in X, and ω is any co-vector in X, then there exists a rule for making a real number from V and ω. We introduce the notation ω V IR (1.44) 18

20 to denote this rule. The operation is linear, and so we have ω U + V = ω U + ω V, ω λ V = λ ω V, (1.45) where U and V are any two vectors, and λ is any real number. Just as one expands vectors with respect to some basis E a, namely V = V a E a, so one expands co-vectors with respect to a dual basis, which we shall denote by e a. Thus we write ω = ω a e a. By definition, the basis and its dual satisfy e a E b = δ a b. (1.46) From the linearity of the mapping from X to X, we therefore have that ω V = ω a e a V b E b = ω a V b e a E b = ω a V b δ b a = ω a V a. (1.47) Note that under the change of basis E a given in (1.38), it follows that the dual basis e a must transform inversely, namely e a e a = Λ a b e b, (1.48) so that the defining property (1.46) is preserved for the primed basis and its dual. Correspondingly, the invariance of ω itself under the change of basis requires that its components ω a transform as ω a ω a = Λ a b ω b. (1.49) At every point p in the manifold M we define the cotangent space Tp (M) as the dual of the tangent space T p (M). The cotangent bundle T (M) is then defined as the space of all possible co-vectors at all possible points: T (M) = p M T p (M). (1.50) Like the tangent bundle T (M), the cotangent bundle has dimension 2n, since the manifold M is n-dimensional and there are n linearly independent co-vectors at each point. An example of a co-vector is the differential of a function. Suppose f(x) is a function on M. Its differential, df, is called a differential 1-form. It is also variously known as the differential, the exterior derivative, or the gradient, of f. It is defined by df V = V f (1.51) 19

21 for any vector V. Recall that V f is the directional derivative of f along the vector V. If we work in a coordinate basis then the basis for tangent vectors is i / x i. Correspondingly, the dual basis for co-vectors is dx i. By definition, therefore, we have dx i j = δ i j. (1.52) This all makes sense, and fits with our intuitive notion of taking the coordinate differential of f, namely as can be seen by a simple calculation: df V V f = V i i f df = i f dx i, (1.53) = i f dx i V j j = i f V j dx i j = i f V j δ i j = i f V i. (1.54) In a coordinate basis, a general co-vector or 1-form ω is expressed as ω = ω i dx i. (1.55) As with a vector, the geometrical object ω itself is independent of any specific choice of coordinates, whilst its components ω i will change when one changes coordinate frame. We can calculate how this happens by implementing a change of coordinate system, x i x i = x i (x j ), and applying the chain rule for differentiation: ω = ω i dx i = ω i x i x j x j ω j dx j, (1.56) where in the second line this is simply the definition of what we mean by the components of ω in the primed frame. Thus we read off ω j = xi x j ω i. (1.57) This may be compared with the transformation rule (1.30) for the components of a vector. Of course, if we form the scalar quantity ω V then we have ω V = ω i V j dx i j = ω i V j δ i j = ω i V i, (1.58) and it is an immediate consequence of (1.30), (1.57) and the chain rule that this is independent of the choice of coordinates, as befits a scalar quantity: ω i V i = x j x i x i x k ω j V k = xj x k ω j V k = δ j k ω j V k = ω j V j. (1.59) 20

22 1.6 An interlude on vector spaces and tensor products For the sake of completeness, and by way of introduction to the next section, it is perhaps useful to pause here and define a couple of widely-used and important concepts. Let us begin with the idea of a Vector Space. A vector space X is a set that is closed under finite vector addition and under scalar multiplication. In the general case, the elements are members of a field 3 F, in which case X is called a vector space over F. For now, at least, our interest lies in vector spaces over the real numbers. The prototype example of a vector space is IR n, with every element represented by an n-tuplet of real numbers (a 1, a 2,, a n ), where the rule of vector addition is achieved by adding component-wise: (a 1, a 2,..., a n ) + (b 1, b 2,..., b n ) = (a 1 + b 1, a 2 + b 2,..., a n + b n ), (1.60) and scalar multiplication, for example by the real number r, is component-wise: r (a 1, a 2,..., a n ) = (r a 1, r a 2,..., r a n ). (1.61) In general, for any elements A, B and C in the vector space X, and any scalars r and s in the field F, one has the rules: Commutativity: A + B = B + A, Associativity of vector addition: (A + B) + C = A + (B + C), Additive identity: 0 + A = A + 0 = A, Additive inverse: A + ( A) = 0, Associativity of scalar multiplication: r (s A) = (r s) A, Distributivity of scalar sums: (r + s) A = r A + s A, Distributivity of vector sums: r (A + B) = r A + r B, Identity for scalar multiplication: 1 A = A. (1.62) Now, let us turn to tensor products. The Tensor Product of two vector spaces X and Y, denoted by X Y, is again a vector space. It obeys a distributive law, in the sense that if X, Y and Z are vector spaces, then X (Y + Z) = (X Y ) + (X Z). (1.63) 3 A Field is any set of elements that satisfies axioms of addition and multiplication, and is a commutative division algebra. Examples of fields are the real numbers IR, the complex numbers IC, and the rational numbers. By contrast, the integers are not a field, since division of integers by integers does not give the integers. 21

23 If elements of the vector spaces X and Y are denoted by x and y respectively, then the tensor-product vector space X Y is spanned by elements of the form x y. The following rules are satisfied: (x 1 + x 2 ) y = x 1 y + x 2 y, x (y 1 + y 2 ) = x y 1 + x y 2, λ (x y) = (λ x) y = x (λ y), (1.64) where λ is any scalar. Note that 0 y = x 0 = 0. If α i is a basis of vectors for X, and β j is a basis of vectors for Y, then α i β j for all (i, j) gives a basis for X Y. In other words, we can expand any vectors x and y in the vector spaces X and Y in the forms x = i x i α i, y = j y j β j, (1.65) and we can expand any vector z in the tensor-product vector space Z = X Y as z = i,j z ij α i β j. (1.66) Note that if the dimensions of the vector spaces X and Y are p and q, i.e. one needs a set of p basis vectors for X, and a set of q basis vectors for Y, then the tensor product X Y has dimension pq. For example, if we take the tensor product IR p IR q, we get a tensor product vector space of dimension pq that is actually just IR pq. 1.7 Tensors Having introduced the notion of vectors and co-vectors, it is now straightforward to make the generalisation to tensors of arbitrary rank. By this is meant geometrical objects which live in a tensor product space, involving, say, p factors of the tangent space T p (M), and q factors of the cotangent space Tp (M). Such a tensor is said to be of type (p, q), and to have rank (p + q). Suppose T is such a tensor. We can then express it in terms of its components in a coordinate basis as T = T i 1 i pj1 j q i1 i2 ip dx j 1 dx j 2 dx jq. (1.67) With the standard philosophy that the tensor T itself is a geometrical object which exists independently of any choice of frame, we therefore see by comparing with its expansion in a primed coordinate frame, T = T i 1 i pj1 j q i 1 i 2 i p dx j 1 dx j 2 dx j q, (1.68) 22

24 where of course i / x i, that the components will transform according to the rule T i 1 i pj1 j q = x i 1 x k 1 x ip x kq x l1 x j 1 xlq x j q T k 1 k p l1 l q. (1.69) In other words, there is a factor of the type x i for each vector index, just like the transformation for V i in (1.30), and a factor of the type xl for each co-vector index, just like x k x j in the transformation of ω i in (1.57). One can view (1.69) as the defining property of a tensor, or, more precisely, the defining property of a general-coordinate tensor, i.e. a tensor with respect to general coordinate transformations. Namely, we can say that T is a type (p, q) tensor under general-coordinate transformations if and only if its components T i 1 i p j1 j q transform like (1.69) under general coordinate transformations. It is obvious that if T and U are two tensors of type (p, q), then T + U is also a tensor of type (p, q). One proves this by the standard technique of showing that the components of T + U transform in the proper way under general coordinate transformations. It is rather obvious that we can take arbitrary products of tensors and thereby obtain new tensors. For example, if V is a (1, 0) tensor (i.e. a vector), and if ω is a (0, 1) tensor (i.e. a co-vector, or 1-form), then W V ω is a tensor of type (1, 1), with components W i j = V i ω j. (1.70) It is clear from the transformation rules (1.30) and (1.57) for V i and ω j that the components W i j transform in the proper way, namely as in (1.69) with p = q = 1. This product is called the Outer Product of V and ω. This terminology signifies that no index contractions are being made, and so the rank of the product tensor is equal to the sum of the ranks of the two tensor factors. In general, we can take the outer product of two tensor of types (p, q) and p, q ), thereby obtaining a tensor of type (p + p, q + q ). Note that the Kronecker delta symbol δj i is nothing but the set of components of a very specific tensor δ of type (1, 1). It is known as an invariant tensor, since it takes the identical form in any coordinate frame. Thus if we take the standard definition of the Kronecker delta in a particular coordinate frame, namely δ i j = 1, if i = j, δ i j = 0, if i j, (1.71) and then compute the components of δ in another coordinate frame, under the assumption that it is a tensor, then from (1.69) we obtain δ i j = δ k l x i x k x l x j = x i x j = δi j, (1.72) 23

25 and so it has the same numerical set of components in all coordinate frames. Another operation that takes tensors into tensors is called Contraction. We can illustrate this with a tensor of type (2, 2); the generalisation to the arbitrary case is immediate. Suppose T is of type (2, 2), with components T ij kl. We can form a tensor of type (1, 1) by contracting, for example, the first upper index and the first lower index: X j l T ij il. (1.73) (Recall that as always, the summation convnetion is operating here, and so the repeated i index is understood to be summed over 1 i n.) The proof that X j l so defined is indeed a tensor is to verify that it transforms properly under general coordinate transformations: X j l T ij il = T mn pq = T mn pq δ p m = X n q x j x n x j x n x q x i x j x m x n x q x p x i x l = T mn mq x q x l x j x n x q x l x l. (1.74) Note that the crucial point is that the transformation matrices for the upper and lower i indices are inverses of one another, and so in the second line we just obtain the Kronecker delta δ p m that implements the contraction of indices on the unprimed tensor T mn pq, giving back X n q. It is clear that the same thing will happen for a contraction of an upper and a lower index in any tensor. A common example of an index contraction, and one which we have in fact already encountered, is in the formation of the so-called Inner Product. If V is a vector and ω is a co-vector or 1-form, then their inner product is given by ω V = ω i V i, (1.75) as in (1.58). This can be viewed as taking the index contraction on their outer product W i j V i ω j defined as in (1.70): W i i = V i ω i. Not surprisingly, since this produces a tensor of type (0, 0) (otherwise known as a scalar), it is invariant under general coordinate transformations, as we saw earlier. Note that one can also perform the operations of symmetrisation or antisymmetrisation of a tensor, and this yields another tensor for which these properties are preserved under general coordinate transformations. For example, if T ij is a general 2-index tensor we can define its symmetric and antisymmetric parts: S ij = 1 2 (T ij + T ji ), A ij = 1 2 (T ij T ji ), (1.76) 24

26 and that T ij = S ij + A ij. It is easy to see that S ij and A ij are both tensors, and that S ij is symmetric in all coordinate frames, and A ij is antisymmetric in all coordinate frames. It is useful to have a notation indicating a symmetrisation or antisymmetrisation over sets of indices. This is done by the use of round or square brackets, respectively. Thus we can rewrite (1.76) as S ij = T (ij) 1 2 (T ij + T ji ), A ij = T [ij] 1 2 (T ij T ji ). (1.77) More generally, symmetrisation and antisymmetrisation over n indices is defined by T (i1 i n) 1 ( ) T i1 i n! n + even permutations + odd permutations, T [i1 i n] 1 ( ) T i1 i n! n + even permutations odd permutations. (1.78) We shall see later that totally antisymmetric tensors of type (0, p) play an especially important role in geometry. They are the p-index generalisation of the co-vector or 1-form, and are known as p-forms. 1.8 The Metric Tensor At this point, we introduce an additional structure on the manifold M, namely the notion of a metric. As its name implies, this is a way of measuring distances in M. It should be emphasised from the outset that there is no unique way of doing this, although very often it may be the case that there is a natural preferred choice of metric (up to scaling), suggested by the symmetries of the problem. Mathematically, we may simply define the metric as a smooth assignment to the tangent space at each point of the manifold a real inner product, or bilinear form, which is linear over functions. We shall also require that this bilinear form be symmetric. Thus if U and V are any vectors, then a metric g is a bilinear map from U and V into the reals g(u, V ) IR, (1.79) with the following properties g(u, V ) = g(v, U), g(λ U, µ V ) = λ µ g(u, V ), (1.80) where λ and µ are arbitrary real numbers. We shall also demand that the metric g be non-degenerate, which means that if g(u, V ) = 0 (1.81) 25